# The space of real places on ℝ(x, y)

### Abstract

The space 𝓜(ℝ(x, y)) of real places on ℝ(x, y) is shown to be path-connected. The possible value groups of these real places are determined and for each one it is shown that the set of real places with that value group is dense in the space. Large collections of subspaces of the space 𝓜(ℝ(x, y)) are constructed such that any two members of such a collection are homeomorphic. A key tool is a homeomorphism between the space of real places on ℝ((x))(y) and a certain space of sequences related to the “signatures” of [2], which themselves are shown here to be related to the “strict systems of polynomial extensions” of [3].

### Keywords

real place; space of real places; strict system of polynomial extensions; Harrison set; path-connected; dense subset

### References

2. Brown R., Valuations, primes and irreducibility in polynomial rings and rational function fields, Trans. Amer. Math. Soc. 174 (1972), 451–488.

3. Brown R., Roots of irreducible polynomials in tame Henselian extension fields, Comm. Algebra 37 (2009), 2169–2183.

4. Brown R., Marshall M., The reduced theory of quadratic forms, Rocky Mountain J. Math 11 (1981), 161–175.

5. Brown R., Merzel J.L., Invariants of defectless irreducible polynomials, J. Algebra Appl. 9 (2010), 603–631.

6. Dubois D.W., Infinite primes and ordered fields, Dissertationes Math. Rozprawy Mat. 69 (1970), 43 pp.

7. Engler A.J., Prestel A., Valued fields, Springer-Verlag, New York, 2005.

8. Harman J., Chains of higher level orderings, in: Dubois D.W., Recio T. (Eds.), Ordered fields and real algebraic geometry, Contemp. Math., 8, Amer. Math. Soc., Providence, 1982, pp. 141–174.

9. Kaplansky I., Maximal fields with valuations, Duke Math. J. 9 (1942), 303–321.

10. Khanduja S.K., Khassa R., On invariants and strict systems of irreducible polynomials over Henselian valued fields, Comm. Algebra 39 (2011), 584–593.

11. Knebusch M., On algebraic curves over real closed fields. I, Math. Z. 150 (1976), 49–70.

12. Kuhlmann K., The structure of spaces of R-places of rational function fields over real closed fields, Rocky Mountain J. Math. 46 (2016), 533–557.

13. Kuhlmann K., Kuhlmann F.-V., Embedding theorems for spaces of R-places of rational function fields and their products, Fund. Math. 218 (2012), 121–149.

14. Lam T.Y., Orderings, valuations and quadratic forms, CBMS Regional Conference Series in Mathematics, 52, Amer. Math. Soc., Providence, 1983.

15. MacLane S., A construction for absolute values in polynomial rings, Trans. Amer. Math. Soc. 40 (1936), 363–395.

16. Rosenstein J.G., Linear Orderings, Pure and Applied Mathematics, 98, Academic Press, New York, 1982.

17. Schülting H.-W., Real points and real places, in: Dubois D.W., Recio T. (Eds.), Ordered fields and real algebraic geometry, Contemp. Math., 8, Amer. Math. Soc., Providence, 1982, pp. 289–295.

*Annales Mathematicae Silesianae*,

*32*, 99-131. Retrieved from https://www.journals.us.edu.pl/index.php/AMSIL/article/view/13916

Department of Mathematics, Univeristy of Hawaii, USA United States

Department of Mathematics, Soka University of America, USA United States

**The Copyright Holders of the submitted text are the Author and the Journal. The Reader is granted the right to use the pdf documents under the provisions of the Creative Commons 4.0 International License: Attribution (CC BY). The user can copy and redistribute the material in any medium or format and remix, transform, and build upon the material for any purpose.**

- License

This journal provides immediate open access to its content under the Creative Commons BY 4.0 license (http://creativecommons.org/licenses/by/4.0/). Authors who publish with this journal retain all copyrights and agree to the terms of the above-mentioned CC BY 4.0 license. - Author’s Warranties

The author warrants that the article is original, written by stated author/s, has not been published before, contains no unlawful statements, does not infringe the rights of others, is subject to copyright that is vested exclusively in the author and free of any third party rights, and that any necessary written permissions to quote from other sources have been obtained by the author/s. - User Rights

Under the Creative Commons Attribution license, the users are free to share (copy, distribute and transmit the contribution) and adapt (remix, transform, and build upon the material) the article for any purpose, provided they attribute the contribution in the manner specified by the author or licensor. - Co-Authorship

If the article was prepared jointly with other authors, the signatory of this form warrants that he/she has been authorized by all co-authors to sign this agreement on their behalf, and agrees to inform his/her co-authors of the terms of this agreement.